This research aims to establish a relationship between the extended complex plane and the
sphere, through a technique that has been studied in ancient times, the stereographic projection.
For this, we will make an introduction to the set of complex numbers, highlighting the most
relevant topics to understand and define the stereographic projection, then we will present its
main properties and two applications, the first relates the stereographic projection and Euclid’s
Pythagorean triple, to second is an application in photography, which shows how an image can
be manipulated to look like a sphere. In addition, this work will address the studies developed
by August Ferdinand Mobius (1790 - 1868), who originated what we know today by M ̈ obius ̈
transformations. After defining these transformations and presenting their main results, we will
show an application involving the transformations in a sphere supported on the complex plane,
and also the elementary Mobius transformations, translation, inversion, and rotation, where we ̈
will use the GeoGebra software to create 3D animations.
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